Record the three-hub extremal actual-reader charge.

[unpaidmoon]

Change and mathematical scope

Scoped proof in one finite row model. No new κ and no selected-frontier change. The selected conditional exponent stays 0.001672660197076918.

On the complete extremal table of the PR #3573 star with h independent hubs, the relaxed literal actual-reader charge exceeds the one-unit baseline by at least 4h(h+1) for every h ≥ 1. For h ≤ 3 the excess is exact:

hubs	rows	baseline	minimum charge	excess
1	20	40	48	8
2	52	104	128	24
3	104	208	256	48

At three hubs, 24 rows cost 2. Twelve of them are crossed five-query readers, two for each pair of hubs. The third hub therefore adds two-bearing rows beyond dxy and gtxy. It does not repeat the twelve-unit K3,2 excess. That twelve is the separate 21-premise certificate, whose charge this package recomputes as 54.

The lower bound is the pairing in PROOF.md, including the determinant -2 that makes each crossed five-query set a minimal reader. Equality through three hubs is the subset census. This is not a graph classification, a characteristic-two transfer, a supplier admission, or a κ improvement.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main; priority revision 6).
• Head: 1391427065829965bf3eb83dac022f4de6c82c48.
• Task: uniform-transfer-audit, scope changed. The card asks for one all-size, compiler, precision, hosting or restoration implication. This session takes the second question of handoff comment 19021149.
• Handoff: Handoffs and requests for independent checks #1166 (comment)

Eligible handoffs, chronological on #1166 from comment 19016211, after dropping later DONE work, this account's delivered round (#4107, #4110–#4120, #4122, #4123), and next tasks already matched by an open PR:

1. Single-hub two-bearing factorization of the PR Classify all thirty two-port support transfers #3573 certificate (comment 19021149).
2. Invoice-level audit of composed T488 ownership (comment 19022532, question 1).
3. Physical rank-8/u2 consumer against the 16 extra rank-one children (comment 19029639).
4. Integer-shear supplier/recurrence inequality left by Prove minimum binary carrier and native XOR bill for the integer-shear target #3947 (comment 19031016).
5. Independent review of the research: certify local-first two-ADD obstruction on staged544 #3942 interaction reduction (reply 19032637).
6. One adjacent completed-call pair (comment 19034066 and reply 19034676).
7. Coupled source-normalization fusion (comment 19035513).
8. Literal-index-to-tensor bridge (comment 19035716).
9. One physical K/adjoint/frame boundary (comment 19036235).
10. Context admission for the encoded radix traversal (comment 19037308).
11. Encoded-boundary review of the many-axis prime producer (comment 19037887).
12. q2 event 58, target local 16 (comment 19039339).
13. Profile of p112-ten-deletion-span census.cpp (comment 19045344).
14. One actual selected stock coefficient (comment 19047276).

Fewer than 29 items remain. Item index (29-1) mod 14 = 0, so the refinement target is item 1. Open PR #4124 already proves that the one-hub charge is 48. This package does not import it and does not repeat that proof as the result. The charge 48 appears only as the h=1 row of the same census. The added statement is the third-hub excess 48 and the separation from the 21-premise charge 54.

Dropped before the cut: the #3605 wrapper (open #3681), the typed actual-width H (#4123), the L22 pair and revisit notes (#4117, #4120), the P104 residual consumer (open #3943), the T385 replay (open #3936), the QQ16 six-tape bind (released, and refined by #4114), neutral-first retiming (open #3941 and #3957), the tensor-pairing review (#4113), the moved-ADD pair (#4122), the separated A2 acceptance (#4112), helpers 600/601 (open #4061), and the R439, batch952, flag-9, V45 and span-four/six notes already delivered this round.

Validation

CPython 3.12.3. Registered replay:

python3.12 -B scripts/research_package.py --package research/p112-third-hub-extremal

Passed locally as PASS_REGISTERED_RESEARCH_PACKAGE research/p112-third-hub-extremal outcome=completed.

• verify.py uses rational row reduction.
• independent.py uses integer cross-multiplication and does not import that file. Both print the same charges.
• test_controls.py checks rank agreement on every two-hub mask, minimality of the one-hub triple, separation of charges 128 and 54, a dropped-row mutation, and the two written minors equal to -2.
• The same commands pass under -O.

Not run: the h=4 census, a characteristic-two transfer, Lean, Docker, hosted CI, and the selected cold replay. Finite enumeration certifies equality only through three hubs. The inequality for every hub count is the written pairing.

Credit

The one-hub star is Nick Petalas's PR #3573. The second hub and the 21 premises are Nick Petalas's PR #3583. The relaxed row model and the charge 54 are Nick Petalas's PR #3698, retaining Philippe Van Velzen's expanded identity, ctoth's coverage question and the actual-reader lineage. The one-hub charge 48, proved separately, is open PR #4124. The third-hub census is by unpaidmoon, with substantial AI assistance (Cursor Grok 4.7). No independent human or proof-kernel review is claimed. Apache-2.0.

Open next questions

1. Does the excess stay equal to 4h(h+1) for every hub count? The inequality is proved. Equality needs the extremal table. The first step is the same census at h=4 (2^15 query masks). Nothing mathematical blocks it; this session stops at the third hub named in the handoff.
2. Does any three-hub extremal row survive a characteristic-two support transfer? The first step is one crossed five-query reader against the shape of PR Classify all thirty two-port support transfers #3573's GF4 map. PR Prove rooted-minor closure of marked support-transfer obstructions #3591 moves support obstructions, not this row cost, and a failed transfer would not by itself raise the pair floor.

Workflow feedback (optional)

The handoff named the third hub as the next comparison after the single-hub charge, which kept this session off the charge-48 proof already opened in #4124. #1200